A Dynamic Model of California's Hardwood
Rangelands1
Richard B. Standiford
Richard E. Howitt2
Abstract: Low profitability of hardwood rangeland manage­
ment, and oak tree harvesting for firewood markets and forage
enhancement has led to concern about the long-term sustainability
of the oak resource on rangelands. New markets for recreational
hunting may give value to oaks for the habitat they provide for
game species, and broaden the economic base for managers. A
ranch level optimal control model has been developed to assess
optimal oak tree canopy and livestock stocking under different
biological and economic conditions. The impact of recreational
hunting on management was also assessed. In general, hunting
improved the economic return on hardwood rangelands, and
resulted in lower oak harvest levels and lower livestock stock­
ing. The policy implication is that public concern over oak tree
harvesting may be partially alleviated by new markets for
recreational hunting on hardwood rangelands.
California's oak-covered rangelands occupy 7.4 million
acres (California Department of Forestry and Fire Protection
1988). Over eighty percent of the area is privately-owned,
providing one-third of the forage for the state's livestock indus­
try. The public goods supplied by hardwood rangelands, including
wildlife habitat, watershed protection and aesthetics, have led to
strong public interest in the management of these private lands.
One of the important ecological characteristics of this area is the
interconnectedness of large expanses of woodland.
Conversion of hardwood rangelands to urban or intensive
agricultural use is one of the leading sources of habitat loss and
fragmentation, amounting to around seven percent of the oak
woodland acreage between 1950 and 1980 (Bolsinger 1988).
Oak clearing for forage enhancement was at one time a major
source of habitat loss, however this practice peaked in the 1950's
and 60's (George 1987). Rapid increases in firewood prices in
the 1970's (Doak and Stewart 1986) suddenly gave value to the
oak trees on rangelands, resulting in increased firewood harvest.
Poor oak regeneration has also been documented in some areas,
leading to some concern about the long-term sustainability of
this resource (Muick and Bartolome 1987).
These concerns about the future of this valuable resource
have resulted in zoning restrictions and ordinances at the local
level, and proposals for statewide regulation of tree removal.
However, no empirical estimates have been made of the effect
of current and future economic conditions on landowner man1
Presented at the Symposium on Oak Woodlands and Hardwood Rangeland
Management, October 31 - November 2, 1990, Davis, California.
2
Forest Management Specialist, Dept. of Forestry and Resource Management,
University of California, Berkeley; Professor, Dept. of Agricultural Eco­
nomics, University of California, Davis.
USDA Forest Service Gen. Tech. Rep. PSW-126. 1991
agement decisions for these large expanses of private land.
Private hardwood rangeland profitability directly affects the rate
of fragmentation due to subdivision. The forces impacting oak
tree harvest, such as income from firewood and increased forage
for livestock, affect the structure of the tree canopy and its value
for different wildlife species. New markets for fee hunting on
private rangelands have recently developed in the state. Since
many of the game species demanded by hunters are enhanced by
oak cover, this new market may serve to provide a market-based
oak conservation incentive for landowners.
STUDY DESIGN
The objective of this study is to assess the likely impacts of
different biological and economic conditions on oak stands by
developing a multiple resource management model for hardwood
rangelands. This requires the development of production func­
tions of the basic processes that impact management decisions
on hardwood rangelands and reflect the relationship between
these resources. In addition, economic models that represent the
behavior of hardwood rangeland owners need to be determined.
Figure 1 shows a schematic flow diagram for multiple
resource decision-making. This assumes that ranchers make
decisions about their level of oak tree retention and cattle
stocking based on cattle and firewood markets, the relationship
between oak tree cover and forage production, the rate of growth
of these resources, and the potential for alternative economic
enterprises such as commercial hunting. As this shows, deci­
sions are made annually based on the biological and economic
factors of these resource values.
To evaluate this decision-making framework, an approach
which addresses both dynamics and interrelated resource values
is needed. Traditional methods of analysis in range economics,
which often focus on forage as an annual crop with partial
budgeting used to compare the costs of range improvements to
livestock gains, were felt to be inadequate for this purpose.
Optimal control theory provides a powerful tool which can be
used to develop the decision rules to determine the optimum
pathway of the stocks of oak trees and cattle over time. The
pathway is "controlled" by the manager through decision vari­
ables, or control variables, that are linked with the tree and cattle
capital stock in the system. The amount of oak firewood cut and
livestock sold (or bought) are the control variables. The ranch
manager makes annual decisions about the level of these control
variables.
279
Figure 1—Decision-making flow diagram for hardwood range management.
Equation (1) below shows the general framework used to
evaluate management decisions on hardwood rangelands. This is
based on a manager maximizing net present value (NPV) over a
time period of T years. This shows the decision to be based on
firewood revenue, livestock revenue, and hunting revenue.
Biological growth functions, known as equations of motion,
provide links between years for the capital stocks. The hypoth­
esized interrelationships between the various hardwood range
resources are shown below.
(1)
Maximize: NPV=
T
∑ DF*t (WRt (WDSELt)+HRt (WDt, HRDt, exog.)+
t=1
LRt (HRDt, CSt, FORt (WDt, exog.)))
such that:
[ Equation of motion
WDt+1= F(WDt, exog.)-WDSELt
for oaks]
HRDt+1= G(HRDt, exog.)-CSt
[ Equation of motion
for livestock]
WD0= INITWD
[ Initial stock of
wood]
HRD0= INITHRD
[ Initial stock of
livestock]
WDSELt ≥ 0
[ Wood cutting
nonnegativity
constraint]
280
where:
DFt= discount factor at time t
WDSELt = volume of firewood cut in time t
WRt (WDSELt)= firewood revenue as a function of
firewood cut in time t
WDt = Stock of oak trees at time t
HRDt= stock of livestock at time t
HRt (WDt, HRDt, exog.) = Hunting revenue at time t as a
function o" the stock of trees and livestock and
exogenous variables (location, wildlife population,
etc.)
CSt= Vector of different classes of livestock sold
LRt (HRDt, CSt FORt (WDt, exog.))= Livestock revenue
at time t as a function of the stock of livestock, forage
production, and the number of livestock sold.
FORt (WDt, exog.)= Forage production expressed as a
function of tree canopy and exogenous range produc­
tivity factors
F(WDt, exog.) = Tree growth during time t as a function
of the stock of oaks and exogenous site factors
G(HRDt) = Livestock growth during time t as a function
of the stock of livestock and exogenous factors (i.e.
cattle breed)
USDA Forest Service Gen. Tech. Rep. PSW-126. 1991
ESTIMATION OF PRODUCTION
FUNCTIONS
Production functions were determined for the various com­
ponents of the hardwood range decision-making model. Com­
plete details on the analysis are in Standiford (1989).
Oak Tree Growth
The dynamic optimization process requires that oak growth
can be assessed for differing stand densities in order to evaluate
the impact of selective oak harvesting on future stand structure
and its impacts on livestock, recreation, and firewood enterprises.
Eighty-one study sites were selected in seven locations in stands
of pure blue oak (Quercus douglasii), mixed blue oak and in­
terior live oak (Q. wislizenii) stands in the Sierra Nevada foothills, and mixed blue oak and coast live oak (Q. agrifolia) stands
in the coastal foothills. Individual tree diameter at breast height
(DBH), total height, crown diameter, and five- and ten-year
radial growth was measured on 972 individual trees. This was
used to develop relationships for oak site index, periodic oak tree
growth, and crown cover-volume relationships (Standiford and
Howitt 1988).
Forage Production
A data set of 142 observations of forage production in the
open and under several different densities of oak canopy was
collected from four different studies, representing time series
ranging from 2 to 22 years (Jensen 1987; McClaren and Bartolome
1989; Kay 1986; Heady and Pitt 1979). Seasonal rainfall and
accumulated annual degree days were included for each study
site from weather records. Combined cross-section and time
series analysis was used to evaluate variability between sites and
years. A seasonal forage yield model was estimated based on
overstory oak cover, accumulated julian days, and accumulated
seasonal rainfall. In this model, oak canopy has a greater effect
in depressing yield on higher rainfall areas, which is consistent
with the results reported in the literature (McClaren and Bartolome
1989).
Hunting Production
A random sample of 60 ranches identified as operating
recreational hunting programs was surveyed, providing data on
the type of hunting lease, price of the hunting lease, number of
hunters, hunter success, location of the club, wildlife habitat at
USDA Forest Service Gen. Tech. Rep. PSW-126. 1991
the club, and detailed cost summaries. The major game species
of interest in these surveys were deer, wild turkeys, and wild
pigs. Revenue and cost functions were estimated on a per acre
basis using hedonic regression (Rosen 1974) to decompose the
costs and revenues to the various physical and biological attributes of the hunting club. Since deer hunting was found on 49
of the 55 usable surveys, analysis was restricted to areas with
deer hunting, with the value from pig and turkey hunting that
operated jointly on deer areas evaluated.
Oak crown cover, a numerical rating for scenery, the
percent of high income hunters, percent of trophy deer har­
vested, the difference in animal unit months (AUMs) with and
without hunting, percent of the hunt club where pig hunting is
allowed, and the expenditure for advertising were all positive
variables in the hunting revenue function. Acreage had a sig­
nificant negative sign in the hunting revenue function, which
shows that as ranch size increases, net revenues per acre decreases
due to more dispersed, extensive type of hunting operations.
Family labor constraints are also more likely to be binding on
large ranches. Distance to several large cities, availability of
cabins, guide services, camping, hunting dogs, and vehicles for
hunter use were not significant in explaining variability in
hunting revenue.
The cost function per acre for hunting clubs was also
evaluated. Advertising cost per acre, a dummy variable for guide
service, and a dummy variable for whether deer tags are pro­
vided, were all highly significant variables. Care of game, hunt
club acreage, transportation services, availability of cabins, and
percent of high income hunters, had very low significance in the
analysis.
Price Expectations
One of the largest factors impacting hardwood range
management is the large year-to-year variability in livestock and
hay prices. Price data for different classes of cattle was collected
(U.S. Department of Agriculture, various years; and California
Department of Food and Agriculture, various years). Fair qual­
ity alfalfa hay was used as a proxy for hay prices, and collected
for the Petaluma and Madera areas of the state from the FederalState Market News Service (various years). Firewood prices per
cord were collected from data in Doak and Stewart (1986) for
delivered prices in the San Francisco Bay Area. All prices were
deflated to constant 1977 dollars using the Price Receive Index.
Ranchers were hypothesized to base cattle, hay and firewood
price expectations on a weighted average of past observations.
To model this, a Box-Jenkins analysis of the time series data
estimated the coefficients and variances for price expectation
models of cattle, hay and firewood prices. A block diagonal
matrix of variances, with off-diagonal covariances between
livestock classes, was calculated to evaluate risk due to price
variability in the optimal control model. Independence was
assumed between firewood, hay, and livestock prices. Since
hunting revenue is usually based on a long-term lease, zero price
variability was assumed for hunting revenue.
281
Firewood Harvesting Cost
Dammann and Andrews (1979) report on the costs of
harvesting, processing, and transporting firewood in hardwood
stands in New Hampshire. This engineering data was used to
construct a model of firewood harvesting costs based on the
volume of wood sold.
THE OPTIMAL CONTROL MODEL
Based on these production functions and the general opti­
mal control framework shown in equation (1), a discrete time
optimal control model was set up with three seasons within a
year to model annual range forage production, and yearly time
increments for hunting and wood volume production. Complete
details on the actual formulation of the model are in Standiford
(1989).
Preliminary runs of the optimal control model showed the
firewood harvest was either at its minimum (i.e., no wood cut)
or at its maximum (i.e., all the wood is clearcut). Since actual
wood harvest data shows that partial harvest is the typical
method for oak tree harvest (Bolsinger 1988), a procedure to
determine missing "costs" of harvesting firewood was used.
Ranchers value oak trees for the value they add to their ranch for
their own personal aesthetics and for game and nongame wildlife species (Huntsinger and Fortmann 1990) even without a
commercial hunt club. A firewood adjustment cost was determined to reflect the net value of the wood harvest to the rancher
by calibrating the model to actual oak tree harvest data over a
thirteen year period for different hardwood range forest types
(Bolsinger 1988). Inclusion of this firewood adjustment cost
allows policy analysis to be based upon actual behavior.
The livestock enterprise is assumed to be a cow-calf op­
eration, the predominant type on hardwood rangelands in
California (California Department of Forestry and Fire Protec­
tion 1988). Livestock revenue in time t is composed of the sale
of feeder calves (steers and heifers) and the sale of cull cattle.
The price expectation models were used to evaluate livestock
price uncertainties. Costs of the livestock enterprise include
variable costs based on the herd size (Van Riet 1988), and feed
costs based on the amount of feed purchased per season, which
is a control variable estimated in the optimization. Hay price
uncertainty was modelled using the price expectation models
developed.
The seasonal forage model assumes that forage growth on
the annual grassland range occurs only in the first two seasons
(i.e., September 1 through May 31), with the residual forage left
in the summer available as low quality dry forage. Forage
availability in a season is based on climatic variables, the number
of livestock animal unit months (AUMs) in that season and
previous seasons, and the quantity of supplemental feed
282
purchased. Seasonal livestock nutritional requirements in AUMs
were determined exogenously using the program COWFLOW
(Bell 1988) based on livestock weights, target rates of gain,
calving percent, and the percent bulls in the herd.
A terminal value was calculated at the end of the control
period to incorporate the value of future earning stream for the
firewood, hunting and livestock enterprises.
A chance constrained approach was chosen to incorporate
producer price uncertainty into the optimal control model
(Charnes and Cooper 1959). Producers incorporate uncertainty
into their management decision by setting the probability level
that the net cash flow each year is positive based on a combina­
tion of indebtedness and interest rate. The higher the probability
level, the less a producer is able to take a loss, representing a
more risk adverse individual.
SOLUTION TECHNIQUE
The discrete time optimal control model was solved for four
control variables, namely forage allocation to hunting, supplemental feed purchased, the number of cattle to hold off the
market as replacement heifers, and the quantity of firewood sold.
The model was also solved for two state variables, namely the
number of cow-calf pairs, and the standing volume of oak trees.
Optimal controls were calculated for the situation assuming no
price variability (the certainty equivalent case), and for the
chance constrained case. Forage variability was considered by
including the actual time series of seasonal rainfall for the areas
being evaluated.
This system was solved using the GAMS/MINOS system
(Brooke and others 1988) for nonlinear optimization on a
personal computer. The problem was solved over a thirteen year
control period to coincide with the firewood harvest calibration
time period.
A low and high quality range site was included in the
analysis, as well as poor, medium, and good oak site indexes.
Four different initial oak volume levels were evaluated, ranging
from 250 to 1000 cubic feet per acre. The effect of hunting was
evaluated by solving the optimal control model for average and
good quality hunting conditions, and for no hunting. The entire
set of policy runs was solved for both the certainty equivalent
case, and for the chance constrained case. This range of sce­
narios resulted in 144 different optimal solutions.
USDA Forest Service Gen. Tech. Rep. PSW-126. 1991
RESULTS
Effect of Hunting on Total Return
The effect of a broadened market base may improve prof­
itability on hardwood rangelands and help reduce conversions.
Figure 2 shows the net present value for poor and good quality
hardwood range site with 750 cubic feet of oak per acre. On a
poor range site, adding hunting as an enterprise increases NPV
by 144 percent (from $50 to $122 per acre), and makes hunting
the dominant economic enterprise. On a good range site, hunting
increases the NPV by 50 percent (from $154 to $231 per acre),
although cattle production is the dominant economic value on
this site. This figure shows the relatively minor contribution that
firewood harvesting makes to the total economic value of the
operation. On the poor range site with hunting, firewood revenue
is only 4 percent of the NPV, while on the good range site with
hunting, firewood harvesting is only 1.5 percent of the total NPV.
Including risk in the analysis increases the percentage that
hunting and firewood revenues occupy in the portfolio, and
decreases the percentage from the livestock enterprise.
Optimal Wood Harvest Levels
Table 1 shows the cumulative firewood harvest over the 13
year control period for three oak site indexes, four initial oak
volume levels, with and without hunting, and two different range
productivity classes. Slightly less oak firewood harvesting oc­
curs when hunting takes place, especially in stands with 750 to
Figure 2—The effect of hunting, firewood harvest, and cattle production on net present value per acre for
a medium quality oak site and a poor and good range site, with 750 cubic feet of oak per acre.
Table 1— Cumulative firewood harvest in cubic feet per acre over 13 years for the certainty equivalent case.
USDA Forest Service Gen. Tech. Rep. PSW-126. 1991
283
1000 cubic feet per acre. This indicates that the marginal
decrease in hunting revenue due to oak canopy changes is
greater than the marginal revenue from the firewood harvest.
Hunting apparently does provide an incentive for hardwood
range managers to conserve oak trees. The small difference
between ranches with and without hunting may be because the
firewood adjustment cost reflects the fact that even without a
hunting club, ranchers' utility is enhanced by habitat for game
and nongame wildlife, and the aesthetic value of oaks. It is of
special interest that no firewood harvesting occurs on areas with
only 250 cubic feet per acre. The marginal cost of harvesting
firewood exceeds the firewood price at these levels. It is also
worth noting that ranchers are not likely to completely clear their
ranges for forage enhancement because the marginal revenue of
the added forage is less than the marginal cost of cutting trees.
Inclusion of the risk term tends to increase the amount of wood
cut.
Optimum Livestock Density
Figure 3 shows the optimal trajectory for the number of
cow-calf pairs over the 14 year control period on a good range
site with initial conditions of 750 cubic feet of oak per acre and
150 cow-calf pairs per 1000 acres. This trajectory shows that
livestock density decreases as hunting quality increases. This
can be thought of as an allocation of forage to wildlife, and also
allocation of management effort to the hunting enterprise. The
allocation of AUMs to either the hunting operation or the
livestock operation are set so that marginal hunting revenue
from the last AUM added to the hunting enterprise equals the
marginal decrease in livestock revenue. When risk is included,
livestock numbers do not show the same degree of annual
fluctuations as the certainty equivalent case.
Purchase of Supplemental Feed
Given the range of hay prices and price variability included
in these scenarios, very little hay is purchased to provide
supplemental feeding for cattle. The optimal control model
shows that only a small amount of hay is purchased and only in
2 out of 13 years. Somewhat more hay is purchased in the
certainty equivalent case which suggests that hay price variabil­
ity has a larger effect than forage variability due to climatic
fluctuations.
GENERAL CONCLUSIONS
This study showed the interrelationship between the stocks
of oak trees and cattle, and how management decisions about
trees harvested, cows bought or sold, hay bought, and the
operation of enterprises such as hunting, all interrelate to affect
ranch profitability. Relatively low oak firewood harvests are
calculated by the model, indicating the low value of firewood and
the fact that ranchers already incorporate nonmarket values
Figure 3—Optimum 13 year trajectory of cow-calf pairs on a good range site with an oak overstory of 750 cubic feet
per acre for three hunting levels, and a certainty equivalent and chance constrained approach to risk.
284
USDA Forest Service Gen. Tech. Rep. PSW-126. 1991
for oaks in their management decisions. Complete oak removal
for forage production appears to be an unlikely management
practice. This study also showed that operation of a commercial
hunting enterprise has a significant effect on ranch profits. In
fact, on poor quality range sites, hunting becomes the dominant
economic production factor. This suggests that diversification
of the ranch portfolio with new economic enterprises may help
decrease subdivision pressure by increasing net profits as well as
spreading out the risk of production. Operation of hunting
enterprises also affects firewood harvest and cattle stocking
levels, again demonstrating the importance of evaluating the
multiple resources that exist on hardwood rangelands.
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